Jesse Ratzkin

Two isoperimetric inequalities for the Sobolev constant

In this note, we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalizes an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.

A numerical investigation of level sets of extremal Sobolev functions

In this paper we investigate the level sets of extremal Sobolev functions. For Ω ⊂ R and 1 ≤ p < 2n n−2 , these functions extremize the ratio ‖∇u‖ L2(Ω) ‖u‖Lp(Ω) . We conjecture that as p increases the extremal functions become more “peaked” (see the introduction below for a more precise statement), and present some numerical evidence to support this conjecture.

On the rate of change of the sharp constant in the Sobolev–Poincaré inequality

We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usual Holder inequality, which may be of independent interest.

Isoperimetric Inequalities for Extremal Sobolev Functions

Let \(\Omega \subset \mathbf{R}^{n}\) be a bounded domain with boundary of class \(\mathcal{C}^{1}\). One can measure various geometric and physical quantities attached to \(\Omega\), such as volume, perimeter, diameter, in-radius, torsional rigidity, and principal frequency. The first chapter of [16] contains a long list of such interesting quantities, as well as their values for standard shapes such as disks, rectangles, strips, and triangles.